Changing a Linked object's Material
When you go to Outliner, right click over Material item and choose Resync Library Override it brings driver to work, but after reopening file it is not working again and I had to Resync again.
From video it looks like you are using 2.92 in this version when saved&amp;reopen (or your Revert), Material became Linked (chain icon) for some reason, that is why driver is not working. It is the same issue like I noted in your previous Q with Action. You have to right click on Material item and manually Add Library Override first (so it became chain with arrow icon). Than Resync LO.

In 2.93 your driver is not working from beginning, even in this version Material has LO icon it is not working anyway and Resync LO is needed as well.
Note: I have zero experiences with LO as mentioned, but it seems to me like a bug or not working feature since LO is very new to blender, so it can be just a corner case yet in development. Try to post it as a bug and mention bug report link here in a comment, so others can follow developers statement of this thing. Thank you

In latest experimental build 2.93 I noticed under User Preferences &gt; Experimental ... there is Override Auto Resync with a link to https://developer.blender.org/T83811 so you can get more info. But seems like not covering your issue yet (from what I tried - still manually).
If someone smarter here can explain more.